On my opinion, this is an ill-defined question: who evaluated an elementary integral for the first time. Euler, who introduced $\exp$ could do this, without giving it the name "Fourier transform". Parseval [4] did a more complicated integral...
–
Alexandre EremenkoNov 23 '13 at 14:05

2

@AlexandreEremenko That's exactly what I'd like to know: Do we have evidence that Euler could do (3)? (It's not that elementary...) Then OK, I replaced "first derived" by "first published". Note that Watson or Encykl. Math. Wiss. attribute scores of formulas -- but not this one, as far as I could find.
–
Francois ZieglerNov 23 '13 at 15:32

@NoamD.Elkies I agree that the formula is elementary modulo Archimedes, I'd just be surprised to see Poisson state it (below, item 2.) without at least as much of an argument as you have given -- if this was the first time it appeared. Instead, he justifies it by calling it a "known formula". To me this suggests there may be an earlier occurrence, and this "first" is what I'm looking for. But maybe it doesn't exist.
–
Francois ZieglerDec 1 '13 at 2:11